My first reaction to the 'cracked' part was 'septarian nodule'. But the pieces are a bit too regular...

I wonder if they are fish scales. They are hexagonal, whereas many fish scales are quadrilateral, but nonetheless...

The other part looks rather like gills - which I have never seen fossilized.

We need a fossil fish expert to comment, please...!

Update: no, I can't really believe those are fish scales. The pebble is clearly well-worn and those 'pieces' are therefore too 3-dimensional to be fish scales. The whole specimen must be a septarian nodule, with the fossil (the gill-like part) forming a nucleus.

Added a third pic to original post trying to show a close up of the corner. Not sure where in Dorset I was just told that the area they found it was well known for fossil finds at that time. The bone like shards feather off at some of the edges.

Thanks. That does shed some light on the shape of the pale bits in 3D. Seeing the dark dividers in both sections initially forced me to think of the pale bits more as tentacles than blades, but some of the exposure on the corner doesn't quite fit with that; it suggests the structure happens to have folded round (probably during burial) - so the pale bits are still at least somewhat blade-like.

The pale bits are not shards or fragments of something more-solid when alive; they are probably biological structures - looking at the gentle wiggles near the tips and how the 'bases' seem to originate from just one side.

They are not invertebrate legs - there is no segmentation.

Tentacles and tube-feet tend not to be blade-like, and have low preservation potential (as do gills).

We can't tell from the photo whether the pale mineral is primary (fossilized bone) or secondary (mineral filling void left by decayed organ). Observation with a hand lens or microscope might just reveal some biological structure in the pale mineral, but it is very unlikely.

I would definitely like a vertebrate palaeontologist to cast an opinion, please...